What is the general solution for 3tan(x)+cot(x)-5csc(x)=0 ?

The general solution for 3tan(x)+cot(x)-5csc(x)=0 is x= pi/3+2pin,x=(5pi)/3+2pin

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3tan(x)+cot(x)-5csc(x)=0

3 tan (x) + cot (x) - 5 csc (x) = 0

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Solution steps

3 tan (x) + cot (x) - 5 csc (x) = 0

Express with sin, cos

\frac{( - 5 + cos ( x ) ) cos ( x ) + 3 sin ^{2} ( x )}{cos ( x ) sin ( x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

(- 5 + cos (x)) cos (x) + 3 sin^{2} (x) = 0

Rewrite using trig identities

3 - 2 cos^{2} (x) - 5 cos (x) = 0

Solve by substitution

cos (x) = - 3, cos (x) = \frac{1}{2}

cos (x) = - 3 :

No Solution

cos (x) = \frac{1}{2} : x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Combine all the solutions

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

3 cos^{2} (x) = 1 - 5 sin (x)

x, 2 sin (3 x) - 1 = 0

2 (cos (t)) 2 - cos (t) - 1 = 0

x, y^{(i v)} - 16 y = - 8 cosh (2 x)

cos (c) = cos (7) cos (6 0^{\circ})

What is the general solution for 3tan(x)+cot(x)-5csc(x)=0 ?
The general solution for 3tan(x)+cot(x)-5csc(x)=0 is x= pi/3+2pin,x=(5pi)/3+2pin